A chaotic system with only one stable equilibrium

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• 作者：Xiong  ; Wang  ; ;   ; ^{wangxiong8686@gmail.com} ; Guanrong  ; Chen
• 关键词：Chaotic attractor ; Stable equilibrium ; &Scaron ; i&rsquo ; lnikov criterion
• 刊名：Communications in Nonlinear Science and Numerical Simulation
• 出版年：2012
• 出版时间：March 2012
• 年：2012
• 卷：17
• 期：3
• 页码：1264-1272
• 全文大小：1077 K

文摘

If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is actually chaotic? Although chaos theory for three-dimensional autonomous systems has been intensively and extensively studied since the time of Lorenz in the 1960s, and the theory has become quite mature today, it seems that no one would anticipate a possibility of finding a three-dimensional autonomous quadratic chaotic system with only one stable equilibrium. The discovery of the new system, to be reported in this Letter, is indeed striking because for a three-dimensional autonomous quadratic system with a single stable node-focus equilibrium, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Although the equilibrium is changed from an unstable saddle-focus to a stable node-focus, therefore the familiar ?i¡¯lnikov homoclinic criterion is not applicable, it is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous broad frequency spectrum, and a period-doubling route to chaos.